Question

3. Given a CRC code with C(7,4), what is the codeword if the data is 1101 and the divisor is given by X^3+1. Show your steps.

Answer 1

To calculate the CRC codeword for C(7,4) with data 1101 and divisor X^3+1, one must perform binary polynomial division using the XOR operation, appending the resulting CRC to the original data to form the codeword.

Step-by-step explanation:

To find the codeword for a CRC code given C(7,4), the data 1101, and the divisor X^3+1, follow these steps:

1. First, multiply the data by x to the power of the degree of the generator polynomial (which is 3 in this case) to make space for the CRC. This means we represent the data as 1101 followed by three zeros: 1101000.
2. Next, we divide this number by the generator polynomial. The polynomial X^3+1 can be represented in binary as 1001.
3. Now we perform polynomial division. The division process in CRC is similar to long division but using XOR instead of subtraction:
4. Divide the first four bits of the augmented data (1101) by the divisor (1001) to get the first part of the quotient, the rest are the bits for the remainder. Keep in mind that in binary division, we XOR the bits instead of subtracting.
5. Write the remainder below the portion of the dividend that has been divided, and bring down the next bit from the dividend.
6. Repeat the process until all bits have been brought down. The remainder of this division is the CRC.
7. final codeword.

Note: Each step of the XOR operation should be carried out carefully, as any mistakes will affect the outcome. Also, remember that there's no borrow or carry over in XOR operations in binary division.